A short-range twin-engine Let L-410 Turbolet transport aircraft, the aircraft belonged to a flying club in the city of Menzelinsk, the TASS news agency reported.
Cosmonauts use the area for training and the flying club has hosted local, European and world championships, club manager Ravil Nurmukhametov said as quoted by TASS.
The state-run cosmonaut training center has suspended ties with the flying club pending an investigation, TASS said citing a source.
RIA quoted local authorities as saying one of the engines could have broken down.
Photographs of some of the paratroopers on board posing in a kit or with a plane were shown on REN TV and on social media.
Russian aviation safety standards have been tightened in recent years, but accidents continue to occur, especially in remote areas.
An aging Antonov An-26 transport plane crashed in Russia’s Far East last month, killing six people. All 28 people aboard an Antonov An-26 turboprop twin engine died in an accident in Kamchatka in July.